Group #15756709 — subgroups

Subgroups — #15756709

Order 3328.

438 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114995	C₂	1	2	yes	maximal
2	#13114993	C₂	1	1	yes	maximal, characteristic
3	#13114963	C₂	1	2	yes	maximal
4	C₂×#11718047	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#107	—	2	2	no	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#384	C₂²	1	2	yes	—
9	C₂×#88057	—	2	2	no	—
10	C₂×#88047	—	2	4	no	—
11	#11718596	C₂²	1	1	yes	characteristic
12	#11718596	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	2	yes	—
14	#11718121	—	2	4	no	—
15	#11718047	C₄	1	4	yes	complemented
16	C₂²×C₈×C₁₃	D₄	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
18	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
19	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
20	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₃	—	2	1	no	—
22	C₂×C₄⋊Dic₁₃	—	2	1	no	—
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂²×#1055	D₄	1	4	yes	—
26	C₁₃×#107	—	4	4	no	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	Q₈	1	2	yes	—
29	#88057	—	4	4	no	—
30	#88047	—	4	8	no	—
31	C₄×#1055	C₂×C₄	1	4	yes	—
32	C₄×#1055	Q₈	1	4	yes	—
33	#8820	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	—	4	2	no	—
35	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₁₃	—	4	1	no	—
38	C₂²×Dic₁₃	—	4	1	no	—
39	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
40	C₂×C₈×C₁₃	D₈	1	4	yes	—
41	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
42	C₁₃×C₄⋊C₄	—	4	4	no	—
43	C₁₃×C₄⋊C₄	—	4	4	no	—
44	C₁₃×C₄⋊C₄	—	4	4	no	—
45	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
48	C₄⋊Dic₁₃	—	4	2	no	—
49	C₄⋊Dic₁₃	—	4	2	no	—
50	C₄⋊Dic₁₃	—	4	4	no	—
51	C₄⋊Dic₁₃	—	4	4	no	—
52	C₂×#1055	—	2	4	no	—
53	C₂×#1055	—	2	4	no	—
54	C₂×#1055	C₄⋊C₄	1	8	yes	—
55	#1845	—	13	2	no	—
56	#1845	D₁₃	1	1	yes	characteristic
57	#1844	—	13	1	no	—
58	#1834	—	13	2	no	—
59	C₂×#322	—	13	1	no	—
60	C₂³×C₁₃	#127	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	8	2	no	—
62	C₂×C₄×C₁₃	—	8	2	no	—
63	C₂×C₄×C₁₃	—	8	2	no	—
64	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
66	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
68	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
69	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
70	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
71	C₂×C₄×C₁₃	#137	1	4	yes	—
72	C₂×Dic₁₃	—	8	1	no	—
73	C₂×Dic₁₃	—	8	1	no	—
74	C₂×Dic₁₃	—	8	2	no	—
75	C₂×Dic₁₃	—	8	2	no	—
76	C₈×C₁₃	#107	1	8	yes	complemented
77	#1055	—	2	8	no	—
78	C₂×#107	—	2	2	no	—
79	C₂×#107	—	26	4	no	—
80	C₂×#106	—	26	2	no	—
81	C₂×C₄×C₈	—	13	2	no	—
82	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
83	#384	—	13	1	no	—
84	#384	—	13	1	no	—
85	#384	C₂×D₁₃	1	2	yes	—
86	#340	—	26	4	no	—
87	#322	—	13	4	no	—
88	C₂²×C₁₃	#384	1	1	yes	characteristic
89	C₂²×C₁₃	#493	1	1	yes	characteristic
90	C₂²×C₁₃	#494	1	1	yes	characteristic
91	C₂²×C₁₃	#496	1	2	yes	—
92	C₂²×C₁₃	#497	1	2	yes	—
93	C₄×C₁₃	—	16	2	no	—
94	C₄×C₁₃	C₂×#107	1	2	yes	—
95	C₄×C₁₃	#428	1	4	yes	—
96	C₄×C₁₃	#500	1	2	yes	—
97	C₄×C₁₃	#501	1	4	yes	—
98	Dic₁₃	—	16	1	no	—
99	Dic₁₃	—	16	1	no	—
100	C₂²×C₈	—	13	4	no	—
101	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄⋊C₄	—	26	2	no	—
106	C₂×C₄⋊C₄	—	26	2	no	—
107	C₂×C₄⋊C₄	—	26	1	no	—
108	C₂×C₄⋊C₄	—	26	1	no	—
109	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
110	#107	—	4	4	no	—
111	#107	—	52	8	no	—
112	#106	—	52	4	no	—
113	C₄×C₈	—	13	4	no	—
114	C₄×C₈	—	13	4	no	—
115	C₄×C₈	Dic₂₆	1	2	yes	—
116	C₄×C₈	C₄×D₁₃	1	2	yes	—
117	C₂×C₁₃	#1721	1	1	yes	characteristic
118	C₂×C₁₃	#1722	1	1	yes	characteristic
119	C₂×C₁₃	#1723	1	2	yes	—
120	C₂×C₁₃	#1845	1	1	yes	characteristic
121	C₂×C₁₃	#1847	1	2	yes	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	52	1	no	—
124	C₂²×C₄	—	52	1	no	—
125	C₂²×C₄	D₄×D₁₃	1	2	yes	—
126	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
127	C₂×C₈	—	13	8	no	—
128	C₂×C₈	—	26	4	no	—
129	C₂×C₈	—	26	4	no	—
130	C₂×C₈	#6101	1	4	yes	—
131	C₂×C₈	#6104	1	4	yes	—
132	C₂×C₈	#6107	1	4	yes	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	4	no	—
136	C₄⋊C₄	—	52	4	no	—
137	C₄⋊C₄	—	52	4	no	—
138	C₄⋊C₄	—	52	2	no	—
139	C₄⋊C₄	—	52	2	no	—
140	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
141	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
142	C₄²	Q₈×D₁₃	1	2	yes	—
143	C₁₃	#8820	1	1	yes	characteristic, complemented, Sylow
144	C₂³	#88194	1	1	yes	characteristic
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	—	8	2	no	—
148	C₂×C₄	—	104	1	no	—
149	C₂×C₄	—	104	1	no	—
150	C₂×C₄	—	104	2	no	—
151	C₂×C₄	—	104	2	no	—
152	C₂×C₄	#88142	1	2	yes	—
153	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
154	C₂×C₄	#88165	1	4	yes	—
155	C₂×C₄	#88172	1	4	yes	—
156	C₂×C₄	C₂×#6101	1	1	yes	characteristic
157	C₂×C₄	C₂×#6104	1	2	yes	—
158	C₂×C₄	C₂×#6107	1	2	yes	—
159	C₂×C₄	#88198	1	1	yes	characteristic
160	C₈	—	26	8	no	—
161	C₈	#88047	1	8	yes	complemented
162	C₂²	#11718637	1	1	yes	characteristic
163	C₂²	#11718734	1	1	yes	characteristic
164	C₂²	#11718753	1	1	yes	characteristic
165	C₂²	#11718815	1	2	yes	—
166	C₂²	#11718851	1	2	yes	—
167	C₄	—	16	2	no	—
168	C₄	—	208	1	no	—
169	C₄	—	208	1	no	—
170	C₄	#11718537	1	4	yes	—
171	C₄	#11718542	1	4	yes	—
172	C₄	C₂×#88047	1	2	yes	—
173	C₄	#11718739	1	2	yes	—
174	C₂	#13114503	1	1	yes	characteristic
175	C₂	#13114508	1	1	yes	characteristic
176	C₂	#13114514	1	2	yes	—
177	C₂	#13114995	1	1	yes	characteristic
178	C₂	#13115003	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13114995

C₂

yes

maximal

#13114993

C₂

yes

maximal, characteristic

#13114963

C₂

yes

maximal

C₂×#11718047

C₂

yes

maximal, characteristic

C₁₃×#1845

C₂

yes

maximal, characteristic

C₂×C₁₃×#107

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

—

C₂×#88057

—

C₂×#88047

—

#11718596

C₂²

yes

characteristic

#11718596

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

—

#11718121

—

#11718047

C₄

yes

complemented

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂²×#1055

D₄

yes

—

C₁₃×#107

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

#88057

—

#88047

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

Q₈

yes

—

#8820

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

C₄⋊C₄

yes

—

#1845

—

#1845

D₁₃

yes

—

—

—

C₂³×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

yes

—

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

C₂×D₁₃

yes

—

#340

—

#322

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#497

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#107

yes

—

C₄×C₁₃

#428

yes

—

C₄×C₁₃

#500

yes

—

C₄×C₁₃

#501

yes

—

Dic₁₃

—

Dic₁₃

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂²⋊D₁₃

yes

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

C₂²×D₁₃

yes

characteristic

110

#107

—

111

#107

—

112

#106

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

Dic₂₆

yes

—

116

C₄×C₈

C₄×D₁₃

yes

—

117

C₂×C₁₃

#1721

yes

characteristic

118

C₂×C₁₃

#1722

yes

characteristic

119

C₂×C₁₃

#1723

yes

—

120

C₂×C₁₃

#1845

yes

characteristic

121

C₂×C₁₃

#1847

yes

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

D₄×D₁₃

yes

—

126

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

#6101

yes

—

131

C₂×C₈

#6104

yes

—

132

C₂×C₈

#6107

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄²

C₂×Dic₂₆

yes

characteristic

141

C₄²

C₂×C₄×D₁₃

yes

characteristic

142

C₄²

Q₈×D₁₃

yes

—

143

C₁₃

#8820

yes

characteristic, complemented, Sylow

144

C₂³

#88194

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

104

—

149

C₂×C₄

—

104

—

150

C₂×C₄

—

104

—

151

C₂×C₄

—

104

—

152

C₂×C₄

#88142

yes

—

153

C₂×C₄

C₄⋊C₄×D₁₃

yes

—

154

C₂×C₄

#88165

yes

—

155

C₂×C₄

#88172

yes

—

156

C₂×C₄

C₂×#6101

yes

characteristic

157

C₂×C₄

C₂×#6104

yes

—

158

C₂×C₄

C₂×#6107

yes

—

159

C₂×C₄

#88198

yes

characteristic

160

C₈

—

161

C₈

#88047

yes

complemented

162

C₂²

#11718637

yes

characteristic

163

C₂²

#11718734

yes

characteristic

164

C₂²

#11718753

yes

characteristic

165

C₂²

#11718815

yes

—

166

C₂²

#11718851

yes

—

167

C₄

—

168

C₄

—

208

—

169

C₄

—

208

—

170

C₄

#11718537

yes

—

171

C₄

#11718542

yes

—

172

C₄

C₂×#88047

yes

—

173

C₄

#11718739

yes

—

174

C₂

#13114503

yes

characteristic

175

C₂

#13114508

yes

characteristic

176

C₂

#13114514

yes

—

177

C₂

#13114995

yes

characteristic

178

C₂

#13115003

yes

—

	subgroup	G/H
center	C₂³	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1845	C₂
socle, prime radical	C₂³×C₁₃	#127
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1845	D₁₃
hypocenter	C₁₃	#8820
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#88194

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1845

C₂

socle, prime radical

C₂³×C₁₃

#127

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
15. #11718047	C₄	semidirect
76. C₈×C₁₃	#107	semidirect
143. C₁₃	#8820	semidirect
161. C₈	#88047	semidirect

H ≅ G/N

type

15. #11718047

C₄

semidirect

76. C₈×C₁₃

#107

semidirect

143. C₁₃

#8820

semidirect

161. C₈

#88047

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13114995	2
#13114993	2
#13114963	2
C₂×#11718047	2
C₁₃×#1845	2
#8820	13

subgroup

index

C₁₃×#1845

Subgroups — #15756709

Conjugacy classes of subgroups

Complements

Maximal subgroups